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{\small \it Journal of Osh University ''Differential Equations'', 2026, Volume 1, Issue 1, P. 67--77.}
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\noindent{\small \bf UDC 517.977.1}

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\noindent{\Large \bf Optimal Control Problem for an Impulsive Systems of Functional-Differential Equations with a Nonlinear Function under the Sign of a First-Order Differential}

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\noindent{\bf Mukhametali K. Mamanov}$^*$
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{\small Osh State University,  332, Lenin street, Osh, Kyrgyzstan}
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{\small
{\bf E-mail:} mamanovmuhametali@gmail.com, \: https://orcid.org/0009-0006-3204-7585}

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\noindent{\bf Gofurjon A. Khasanov}
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{\small Samarkand State University, University blv, 15, Samarkand, 140104 Ozbekistan}
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{\small
{\bf E-mail:} khasanov\_g75@mail.ru, \: https://orcid.org/0009-0007-7357-4557}

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\noindent{\bf Zukhra A. Madatova}
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{\small Tashkent University of Information Technology, Tashkent, Ozbekistan}
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{\small
{\bf E-mail:} zuhra\_madatova@mail.ru, \: https://orcid.org/0009-0001-0638-4362}

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\begin{center}
{\bf Received}: November 03, 2025;  {\bf First on-line published}: January 10, 2026
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\parbox{146mm}{\noindent\textbf{Abstract.} \small In this paper the optimal control problems for an impulsive systems of functional-differential equations with a nonlinear function under the sign of the first-order differential and with maxima are investigated. The Initial value problem is reduced to a system of nonlinear functional-integral equations in a Banach space $BD\big([0,T],\mathbb{R}^n\big)$.
In the fixed values of control function, by the method of contracting mappings proves the existence and uniqueness of state function for a nonlinear systems of functional-integral equations with maxima. Then using functional of quality, we are built Pontryagin's function and obtain criteria of optimality. Then we are proved existence and uniqueness of control function.}

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\parbox{146mm}{\noindent\textbf{Key words:} \small \it Optimal control, impulse system with maxima, nonlinear function under the differential sign, rotation of homotopic vector fields, nonzero index of an isolated singular point, existence and uniqueness of a periodic solution.}

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\parbox{146mm}{\noindent\textbf{MSC 2020:} \small 34B10, 34B15, 34B37, 34H05.}


{\centering
\section*{Introduction. Problem statement}}

The theory of functional differential equations has applications in different sciences (see, \cite{chet}--\cite{ka}).
Functional differential equations containing a nonlinear function under the derivative sign arise when solving problems of nonlinear mechanics and nonlinear optimal control. Of theoretical interest is the study of solutions of functional-differential equations when at given points have a discontinuity of the first kind.

There are a lot of publications of studying on differential equations with impulsive effects, describing many
natural and practical processes, are appeared \cite{t1}--\cite{t7}. The interest on the study of nonlocal
problems for the impulsive differential equations is high by the mathematicians \cite{t8}--\cite{t21}.

Optimal control theory constitutes a strong area within the mathematical sciences, offering a wide range of practical applications. Many real-world problems reduce to finding an optimal control function along with the corresponding state function. A wide range of analytical and numerical methods has been developed to solve such problems effectively across disciplines in science and engineering (see, \cite{tt1}--\cite{tt14}).

In this paper problems of optimal control for an impulsive systems of functional-differential equations with a nonlinear function under the sign of the first-order differential are considered. On the interval $[0,T]$ for $t\ne t_{i} ,\, \, i=1,2,...,p$ an impulsive systems of first-order functional-differential equations are considered
\begin{equation} \label{Grind1}
\frac{d}{dt} \Big[x(t)+H(x(t))\Big]=F\left(t,x(t),\max \big\{x(\tau ):\tau \in [t-h,t]\big\},u(t)\right),
\end{equation}
where $0=t_{0} <t_{1} <...<t_{p} <t_{p+1} =T,$ $\,x\in \bar{X},$ $\,\bar{X}$ is closed set in bounded space $\mathbb{R}^{n} $, $F(t,x,y,u)\in C\left([0,T]\times \mathbb{R}^{n} \times \mathbb{R}^{n}\times\Upsilon,\mathbb{R}^{n} \right),$  $H(x)\in PC\left(\mathbb{R}^{n}, \mathbb{R}^{n} \right),$ $\,0<h={\rm const}.$

The system \eqref{Grind1} is studied under following conditions
\begin{equation} \label{Grind2}
x(0)=x_0\neq 0,\, \, \,\; H(x(0))=H(x_0)=H_0\neq 0,
\end{equation}
\begin{equation} \label{Grind3}
x(\xi )=\varphi (\xi),\, \, \,\; \xi \in \big[-h,0^{-}\big]
\end{equation}
and under conditions with given impulses
\begin{equation} \label{Grind4}
x\left(t_{i}^{+} \right)-x\left(t_{i}^{-} \right)=V_{1,i} \left(x\left(t_{i} \right)\right),\, \, \, i=1,2,...,p,
\end{equation}
\begin{equation} \label{Grind5}
H\left(x\left(t_{i}^{+} \right)\right)-H\left(x\left(t_{i}^{-} \right)\right)=V_{2,i} \left(x\left(t_{i} \right)\right),\, \, \,  i=1,2,...,p,
\end{equation}
where $x(t) \in PC\big([0,T],\mathbb{R}^{n} \big)$ is state function, $u(t) \in C\big([0,T],\mathbb{R}^{n} \big)$ is control function, $V_{\kappa} \in C\big(\mathbb{R}^{n},\mathbb{R}^{n} \big),\, \, \kappa =1,2$, $\,x\left(t_{i}^{+} \right)$, $\,x\left(t_{i}^{-} \right)$ are right and left-hand limits.

We use the Banach space $C\big([0,T],\mathbb{R}^{n} \big)$, which contains a vector function $x(t)$, defined and continuous on the segment $[0,T]$, with the norm
\[\left\| \, x(t)\, \right\| =\sqrt{\sum _{j=1}^{n}{\mathop{\max }\limits_{0\le t\le T}} \left|\, x_{j} (t)\, \right| } .\]
Let us also consider the following Banach space
\[PC\left([0,T],\mathbb{R}^{n} \right)=\Big\{x:[0, T]\to \mathbb{R}^{n} ;\, \, x(t)\in C\left(\left(t_{i} ,t_{i+1} \right],\mathbb{R}^{n} \right),\, \, i=1,...,p\Big\},\]
in which $x\left(t_{i}^{+} \right)$ and $x\left(t_{i}^{-} \right)$ $(i=0,1,...,p)$ exist and are bounded, with norm
\[\left\| \, x(t)\, \right\| _{PC[0,T]} =\max \left\{\left\| \, x(t)\, \right\| _{C[t_{i} ,t_{i+1} ]} ,\, \, \, i=1,2,...,p\right\}.\]

We consider a control function
$$u(t) \in \Upsilon=\big\{ u: |\,u(t)\,| \leq M^{*} , \;t \in [0,T],\; 0<M^{*}={\rm const} \big\}$$
and the following functional of quality
\begin{equation}\label{func}
J[u]=\big[x(T)-\alpha \big]^2+\beta \int \limits_{0}^{T} \varepsilon(t) u^2(t)dt,
\end{equation}
where $\alpha, \beta={\rm const}$, $\varepsilon(t) \in C[0,T]$.

\noindent\textbf{Problem statement.} Find a control function $u^{*}(t)\in \Upsilon$ and corresponding values of state function $x^{*}(t)\in PC\big([0, T],\mathbb{R}^{n} \big)$ providing a minimum of functionality \eqref{func} and for all $t\in [0, T],\, \, t\ne t_{i} ,\, \, \, i=1,2,...,p$ state function satisfies the given system \eqref{Grind1}, the conditions \eqref{Grind2} together with the condition \eqref{Grind3}, for $t=t_{i} ,\, \, i=1,2,...,p\, \, \big(0<t_{1} <t_{2} <...<t_{p} <T\big)$ satisfies the impulse conditions \eqref{Grind4} and \eqref{Grind5}.

{\centering
\section{Reduction of the problem \eqref{Grind1}-\eqref{Grind5} to a system of nonlinear functional integral equations}}

Let the function $x(t)\in PC\big([0,T],\mathbb{R}^{n} \big)$ be a solution of the problem \eqref{Grind1}-\eqref{Grind5} for fixed values of control function. Then, taking into account the equalities $x\big(0^{+}\big)=x(0)$, $\,x\big(t_{p+1}^{-} \big)=x(t)$, integrating the system \eqref{Grind1} once in each of the intervals: $\left(0,t_{1} \right]$, $\,\left(t_{1} ,t_{2} \right]$, $\, .\, .\, .\, $, $\,\left(t_{p} ,t_{p+1} \right]$ and combining these integrals, we obtain:
\[\int \limits_{0}^{t}F\left(s,x(s),\max \big\{x(\tau ):\tau \in \left[s-h,s\right] \big\},u(s)\right)ds \]
\[=\left[x\left(t_{1} \right)-x\left(0^{+} \right)\right]+\left[x\left(t_{2} \right)-x\left(t_{1}^{+} \right)\right]+\,.\,.\,.\,+\left[x(t)-x\left(t_{p}^{+} \right)\right]\]
\[+\left[H(x(t_{1} ))-H(x(0^{+} ))\right]+\left[H(x(t_{2} ))-H(x(t_{1}^{+} ))\right]+\,.\,.\,.\,+\left[H(x(t))-H(x(t_{p}^{+} ))\right]\]
\[=-x(0)-\left[x\left(t_{1}^{+} \right)-x\left(t_{1} \right)\right]-\left[x\left(t_{2}^{+} \right)-x\left(t_{2} \right)\right]-\,.\,.\,.\,-\left[x\left(t_{p}^{+} \right)-x\left(t_{p} \right)\right]+x(t)\]
\[-H(x(0))-\left[H(x(t_{1}^{+} ))-H(x(t_{1} ))\right]-\left[H(x(t_{2}^{+} ))-H(x(t_{2} ))\right]\]
\[-\,.\,.\,.\,-\left[H(x(t_{p}^{+} ))-H(x(t_{p} ))\right]+H(x(t)).\]
Taking into account the impulse conditions \eqref{Grind4} and \eqref{Grind5}, we write the last equalities as follows
\[
x(t)+H(x(t))=x(0)+H(x(0))+\sum _{0<t_{i} <t}\Big[V_{1,i} \left(x\left(t_{i} \right)\right)+V_{2,i} \left(x\left(t_{i} \right)\right)\Big]+
\]
\begin{equation} \label{Grind6}
+\int \limits_{0}^{t}F\left(s,x(s),\max \big\{x(\tau ):\tau \in [s-h,s]\big\},u(s)\right)ds.
\end{equation}
For the function $x(t)\in PC\big([0,T],\mathbb{R}^{n} \big)$ in the representation \eqref{Grind6} we will use the conditions \eqref{Grind2} and obtain a system of nonlinear functional-integral equations (SNFIE):
\[
x(t)=x_{0} -H(x(t))+H_{0}+\int \limits_{0}^{t}F\left(s,x(s),\max \big\{x(\tau):\tau \in \left[s-h,s\right]\big\},u(s)\right)ds+
\]
\begin{equation} \label{Grind10}
+\sum _{0<t_{i} <t}\Big[V_{1,i} \left(x\left(t_{i}\right)\right)+V_{2,i} \left(x\left(t_{i} \right)\right)\Big].
\end{equation}
\begin{lemma} \label{LL1} Let be fulfilled the following conditions:\\
$1).$ $M_{F} =\left\| \, F(t,x,y,u)\, \right\| _{C[0,T]} ,\, \, \, 0<M_{F} ={\rm const}$;\\
$2).$ $M_{V_{\kappa}} ={\mathop{\max }\limits_{1\le i\le p}} \left\| \, V_{\kappa ,i}(x)\, \right\| _{C[0,T]} ,\, \, \, 0<M_{V_{\kappa } } ={\rm const},\, \, \kappa =1,2$;\\
$3).$ $\left|\, H\left(x_{1} \right)-H\left(x_{2} \right)\, \right|\le L_{H} \left|\, x_{1} -x_{2} \, \right|,\, \, \, 0<L_{H} ={\rm const}$.

Then for the SNFIE \eqref{Grind10} is true the following estimate
\begin{equation} \label{Grind11}
\left\| \, x(t)-x_{0} \, \right\|_{PC[0,T]} \le \frac{TM_F+p\big(V_1+V_2\big)}{1+L_{H}}.
\end{equation}
\end{lemma}

\begin{proof}[\bf \indent Proof] We rewrite the SNFIE \eqref{Grind10} as
\[
\|\,x(t)-x_{0}\,\|_{PC[0,T]}+\|\,H(x(t))-H(x_{0})\,\|_{PC[0,T]} \leq
\]
\[\leq \int \limits_{0}^{t}\|\,F\left(s,x(s),\max \big\{x(\tau):\tau \in \left[s-h,s\right] \big\},u(s)\right)\,\|_{C[0,T]}ds +\]
\begin{equation} \label{Grind12}
+\sum _{0<t_{i}<t}\Big[\|\,V_{1,i} \left(x\left(t_{i} \right)\right)\,\|+\|\,V_{2,i} \left(x\left(t_{i} \right)\right)\,\|\Big].
\end{equation}

From \eqref{Grind12} we and obtain that
\[\big(1+L_{H} \big)\left\| \, x(t)-x_{0} \, \right\| _{PC[0,T]}\leq TM_F+p\big(V_1+V_2\big).\]
Hence, we get \eqref{Grind11}. Lemma \ref{LL1} is proved. \end{proof}

\begin{lemma}[\cite{Yuldashev3}] \label{LL2} For the difference of two functions with maxima there holds the following estimate
\[
\left\| \, \max \big\{x(\tau ):\tau \in [t-h,t]\big\}-\max \big\{y(\tau ):\tau \in [t-h,t]\big\}\, \right\| _{C[0,T]}
\]
\[
\le \left\| \, x(t)-y(t)\, \right\| _{C[0,T]} +h\left\| \, x'(t)-y'(t)\, \right\| _{C[0,T]},
\]
where $0<h={\rm const}$.
\end{lemma}

{\centering
\section{Unique solvability of the initial value problem}}

By the $BD\left([0,T],\mathbb{R}^{n} \right)$ we denote the Banach space on the interval $[0,T]$ with the norm
\[\left\| \, x(t)\, \right\| _{BD[0,T]} \le \left\| \, x(t)\, \right\| _{PC[0,T]} +h\, \left\| \, x'(t)\, \right\| _{PC[0,T]},\]
$0<h={\rm const}$.

For fixed values of the control function $u(t)$ we proof the following theorem.
\begin{theorem} \label{TT1} Suppose that the conditions of Lemma \ref{LL1} and the following conditions are satisfied:\\
$1).$ $\varphi (\xi )\in C[-h,0]$;\\
$2).$ $M_{H} =\max \limits_{0\le t\le T} \left|\, H\left(x\right)\, \right|<\infty ,\, \, \, \, M_{H'} =\max \limits_{0\le t\le T} \left|\, \frac{d}{dt}H\left(x(t)\right)\, \right|<\infty $;\\
$3).$ For all $t\in [0,T]$, $x,y \in \mathbb{R}^{n} $ there exist functions $\,0<L_{\kappa ,F} (t)\in C\left([0,T],\mathbb{R}^{n} \right),\, \, \, \, \, \kappa =1,2,$ such that
\[\left|F(t,x_{1} ,y_{1} )-F(t,x_{2} ,y_{2} )\right|\le L_{1,F} (t)\, \left|\, x_{1} -x_{2} \, \right|+L_{2,F} (t)\, \left|\, y_{1} -y_{2} \, \right|;\]
$4).$ For all $x\in \mathbb{R}^{n} ,\, \, i=0,1,...,p$ there exist $0<L_{V_{\kappa ,i}} ={\rm const},\, \, \, \kappa =1,2,$ such that
\[\left|\, V_{\kappa ,i} (x_{1} )-V_{\kappa ,i} (x_{2} )\, \right|\le L_{V_{\kappa ,i}} \left|\, x_{1} -x_{2} \, \right|;\]
$5).$ For all $x\in \mathbb{R}^{n}$, there exists $0<L_{H'} ={\rm const},$ such that
\[\left|\, \frac{d}{dt}\Big[H (x_{1})-H (x_{2})\Big]\, \right|\le L_{H'} \left|\, x_{1} -x_{2} \, \right|;\]
$6).$ The radius of the inscribed ball in $X$ is greater than $ \frac{TM_{F} }{1+L_{H}} +p\frac{M_{V_{1} } +M_{V_{2}}}{1+L_{H}};$\\
$7).$ $\rho =\chi _{11} +h\chi _{21} <1,$ where $\chi _{11} $  and $\chi _{21} $ are determined below from the formulas \eqref{Grind24} and \eqref{Grind27}.

Then the problem \eqref{Grind1}--\eqref{Grind5} for all $t\in [0, T],\, \, t\ne t_{i} , \, \, i=1,2,...,p$ has a unique solution $x(t)\in BD\left([0,T],\mathbb{R}^{n} \right),$ which is found from the system of functional-integral equations \eqref{Grind10}.
\end{theorem}

\begin{proof}[\bf \indent Proof] We will show that the right-hand side of the system of equations \eqref{Grind10} as an operator takes a ball with radius $\frac{TM_{F}}{1+L_{H}} +p\frac{M_{V_{1}} +M_{V_{2}}}{1+L_{H}}$ into itself and is a contraction operator. So, according to the Lemma \ref{LL1}, we have the estimate \eqref{Grind11}:
\[\left\| \, x(t)-x_{0} \, \right\|_{PC[0,T]} \le \frac{TM_{F}}{1+L_{H}} +p\frac{M_{V_{1}} +M_{V_{2}}}{1+L_{H}} .\]
In the future we need the derivative of the system \eqref{Grind10}:
\begin{equation} \label{Grind20}
x'(t)+\frac{d}{dt}H(x(t))=F\big(t,x(t),\max \big\{x(\tau):\tau \in [t-h,t]\big\},u(t)\big).
\end{equation}

Similarly to the Lemma \ref{LL1}, passing to the norm in the equation \eqref{Grind20}, we obtain
\begin{equation} \label{Grind21}
\left\| \, x'(t)\, \right\|_{PC[0,T]} \le \frac{M_{F}}{1+M_{H'}}.
\end{equation}
We consider a difference of two functions $x(t)-y(t),$ where the functions $x(t)$ and $y(t)$ are satisfied the system of equations \eqref{Grind10}. By the conditions of the theorem, from \eqref{Grind10} we have
\[
\left|\, x\left(t\right)-y\left(t\right)\, \right|+ L_{H} \left|\, x\left(t \right)-y\left(t \right)\, \right| \leq
\int \limits_{0}^{t}\Big\{L_{1,F} (s)\left|\, x\left(s \right)-y\left(s \right)\, \right|+
\]
\[ +L_{2,F} (s)\left|\, \max \big\{x(\tau ):\tau \in [s-h,s] \big\}-\max \big\{y(\tau):\tau \in [s-h,s] \big\}\, \right|\Big\}ds+\]
\begin{equation} \label{Grind22}
+\sum _{i=1}^{p}\Big[L_{V_{1,i}} +L_{V_{2,i}} \Big]\big|\, x\left(t_{i} \right)-y\left(t_{i} \right)\, \big|.
\end{equation}
Passing to the norm and applying Lemma \ref{LL2} to the inequality \eqref{Grind22}, we have
\[
\left\| \, x(t)-y(t)\, \right\| _{PC[0,T]} + L_{H} \left\| \, x(t)-y(t)\, \right\| _{PC[0,T]} \leq
\]
\[ \leq T\left[\left(\left\| \, L_{1,F} (t)\, \right\|_{C[0,T]} +\left\| \, L_{2,F} (t)\, \right\|_{C[0,T]} \right)\left\| \, x(t)-y(t,x)\, \right\|_{PC[0,T]}+ \right. \]
\[\left. +h\, \left\| \, L_{2,F} (t)\, \right\|_{C[0,T]} \, \left\| \, x'(t )-y'(t )\, \right\| _{PC} \right]+\]
\begin{equation} \label{Grind23}
+\sum _{i=1}^{p}\left[L_{V_{1,i}} +L_{V_{2,i}} \right]\left\| \, x\left(t \right)-y\left(t \right)\, \right\|_{PC[0,T]} .
\end{equation}
If in \eqref{Grind23} we make designations
\begin{equation} \label{Grind24}
\chi _{11} =\frac{T \left(\left\| \, L_{1,F} (t)\, \right\| _{C[0,T]} +\left\| \, L_{2,F} (t)\, \right\|_{C[0,T]} \right)+\sum _{i=1}^{p}\left[L_{V_{1,i}} +L_{V_{2,i}} \right]}{1+L_{H}},
\end{equation}
\[
\chi _{12} = \frac{T\left\| \, L_{2,F} (t)\, \right\|_{C[0,T]}}{1+L_{H}},
\]
then, by virtue of $\chi _{11}>\chi _{12}$, from \eqref{Grind23} obtain
\begin{equation} \label{Grind25}
\left\| \, x(t)-y(t)\, \right\|_{PC} \leq
\chi _{11} \Big[\left\| \, x(t)-y(t)\, \right\|_{PC} +h \left\| \, x'(t)-y'(t)\, \right\|_{PC} \Big].
\end{equation}

By similar way, according to the assumptions of the theorem, from \eqref{Grind20} we have
\begin{equation} \label{Grind26}
 \left\| \, x'(t)-y'(t)\, \right\| _{PC}
\leq \chi _{21} \left\| \, x(t)-y(t)\, \right\|_{PC} +h\,\chi _{22} \left\| \, x'(t)-y'(t)\, \right\| _{PC},
\end{equation}
where
\begin{equation} \label{Grind27}
\chi _{21} =\frac{\left(\left\| \, L_{1,F} (t)\, \right\|_{C[0,T]} +\left\| \, L_{2,F} (t)\, \right\| _{C[0,T]} \right)
+ \sum _{i=1}^{p}\left[L_{V_{1,i}} +L_{V_{2,i}} \right]}{1+L_{H'}},
\end{equation}
\[\chi _{22} =\frac{\left\| \, L_{2,F} (t)\, \right\| _{C[0,T]}}{1+L_{H'}}.\]
Since $\chi _{21}>\chi _{22}$, multiplying both sides of \eqref{Grind26} to positive constant $h$ and the result adding to \eqref{Grind25}, we obtain
\begin{equation} \label{Grind28}
 \left\| \, x(t)-y(t)\, \right\| _{BD[0,T]} \le \rho \, \left\| \, x(t)-y(t)\, \right\| _{BD[0,T]},
\end{equation}
where $\rho =\chi _{11} +h\,\chi _{21}.$

According to the last condition of the theorem, $\rho <1.$ So, from the estimate \eqref{Grind28} we deduce that the operator on right-hand side of the system \eqref{Grind10} is compressing. From the estimates \eqref{Grind11}, \eqref{Grind21} and \eqref{Grind28} implies that there exists unique fixed point $x(t,x_{0})\in BD\big([0,T], \mathbb{R}^n\big).$ The Theorem \ref{TT1} is proved. \end{proof}

{\centering
\section{Optimal control function}}
	
Let $u^{*} (t)$ be optimal control function:
	\[\Delta \, J \big[u^{*} (t)\big]=J \big[u^{*} (t)+\Delta \, u^{*} (t)\, \big]-J\, \big[u^{*} (t)\big]\ge 0,\]
where  $u^{*} (t)+\Delta \, u^{*} (t)\in C[0,T].$
	
Let us consider equation \eqref{Grind10} and the problem of finding the control function
\[u^{*} (t)\in \Upsilon=\Big\{ u^{*} :\, \left|\, u^{*} (t)\, \right|\le M^{*}_{0},\, \, t\in [0,T] \Big\}\]
and the corresponding state function $x^{*}(t)$ that minimize the functional \eqref{func}.

Let's construct the Pontryagin function:
\[H\left(u (t),x (t)\right)=\psi (t)\Big[ F\left(t,x(t),\max \big\{x(\tau):\tau \in \left[t-h,t\right] \big\},u(t)\right)+\]
\[ +V_{1,i} \left(x\left(t_{i}\right)\right)+V_{2,i} \left(x\left(t_{i} \right)\right)\Big]-\beta \varepsilon (t)\, u^{2} (t),\]
where the function $\psi(t)$ will be determined from the equation:
\begin{equation} \label{Grind29}
\dot{\psi} (t)=-\psi (t)
\end{equation}
with condition
\begin{equation} \label{Grind30}
\psi(T)=-2\big[x(T)-\alpha \big].
\end{equation}
Solving this differential equation (\ref{Grind29}), we obtain
\[\psi (t)=C_{0} \exp \{-t \}.\]
To find $C_{0}$ we use the condition (\ref{Grind30})
\[
\psi (T)=C_{0} \exp \{-T \}=-2\bigg[x_{0} -H(x(T))+H_{0}+\int \limits_{0}^{T}F\left(s,x(s),\max \big\{x(\tau):\tau \in [s-h,s]\big\},u(s)\right)ds+\]
\[+\sum _{0<t_{i} <T}\Big[V_{1,i} \left(x\left(t_{i}\right)\right)+V_{2,i} \left(x\left(t_{i} \right)\right)\Big]-\alpha \bigg].\]
Hence, we get
\[\psi (t)=C_{0} \exp \{-t \}=-2\big(x_0+H_0\big)\exp \{T-t \}-2\exp \{T-t \}\bigg[-H(x(T))+\]
\[+\int \limits_{0}^{T}F\left(s,x(s),\max \big\{x(\tau):\tau \in \left[s-h,s\right] \big\},u(s)\right)ds+\]
\begin{equation} \label{Grind31}
+\sum _{0<t_{i} <T}\Big[V_{1,i} \left(x\left(t_{i}\right)\right)+V_{2,i} \left(x\left(t_{i} \right)\right)\Big]-\alpha \bigg].
\end{equation}

The optimality condition is as follows:
\[H_{u}\left(u (t),x (t)\right)=\]
\begin{equation} \label{Grind32}
=\psi (t) F_{u}\left(t,x(t),\max \big\{x(\tau):\tau \in \left[t-h,t\right] \big\},u(t)\right)-2\beta \varepsilon (t)\, u (t)=0.
\end{equation}

We suppose that $\varepsilon (t) \neq 0$. Substituting expression (\ref{Grind31}) into equation (\ref{Grind32}), we obtain an equation with respect to the control function $u (t)$:
\[u (t)=\bigg\{P(t)+Q(t)\bigg[-H(x(T))+\int \limits_{0}^{T}F\left(s,x(s),\max \big\{x(\tau):\tau \in \left[s-h,s\right] \big\},u(s)\right)ds+\]
\begin{equation} \label{Grind33}
+\sum _{0<t_{i} <T}\Big[V_{1,i} \left(x\left(t_{i}\right)\right)+V_{2,i} \left(x\left(t_{i} \right)\right)\Big] \bigg] \bigg\} F_{u}\left(t,x(t),\max \big\{x(\tau):\tau \in \left[t-h,t\right] \big\},u(t)\right),
\end{equation}
where
\[ P(t)=\frac{-2\big(x_0+H_0\big)\exp \{T-t \}}{2\beta \varepsilon (t)}-\alpha Q(t),\:\:Q(t)=\frac{-2\exp \{T-t \}}{2\beta \varepsilon (t)}.\]

For fixed values of state function $x(t)$ we prove that the following theorem is true.

\begin{theorem}\label{TT2} Let the conditions of the theorem \ref{TT1} are fulfilled. If the following conditions hold\\
$1).$ $M_{F_u} =\max \limits_{0\le t\le T} \left|\, F_u\left(t,x,y,u\right)\, \right|<\infty $;\\
$2).$ For all $t\in [0,T]$, $x,y \in \mathbb{R}^{n} $ there exists function $\,0<L_{F} (t)\in C\left([0,T],\mathbb{R}^{n} \right),$ such that
\[\left|\,F(t,x ,y,u_{1})-F(t,x,y,u_{2})\,\right|\le L_{F} (t)\, \left|\, u_{1} -u_{2} \, \right|;\]
$3).$ For all $t\in [0,T]$, $x,y \in \mathbb{R}^{n} $ there exists function $\,0<L_{F_u} (t)\in C\left([0,T],\mathbb{R}^{n} \right),$ such that
\[\left|\,F_u(t,x ,y,u_{1})-F_u(t,x,y,u_{2})\,\right|\le L_{F_u} (t)\, \left|\, u_{1} -u_{2} \, \right|;\]
$4)$. $\rho_1<1$, where \[\rho_1=M_0L_{F_{u}}+TM_{F}L_{F_{u}}+TM_{F_u}L_{F},\:\;M_0=\|\,P(t)\,\|+\|\,Q(t)\,\|\Big[M_H+p\Big(M_{V_{1}}+M_{V_{2}}\Big) \Big].\]

Then the functional equation (\ref{Grind33}) has a unique solution $u(t) \in C\big([0,T],\mathbb{R}^n\big)$.
\end{theorem}

\begin{proof}[\bf \indent Proof] We use the method of contracting mapping in the space $C\big([0,T],\mathbb{R}^n\big)$.
We derive
\begin{equation} \label{Grind34}
\|\,u (t)\,\|\leq \bigg\{\|\,P(t)\,\|+\|\,Q(t)\,\|\Big[M_H
+TM_F+p\Big(M_{V_{1}}+M_{V_{2}}\Big) \Big] \bigg\} M_{F_{u}}<\infty.
\end{equation}

Further, we use the following scheme:
\[|\,f^kg^k-f^{k-1}g^{k-1}\,|\leq |\,f^kg^k-f^{k-1}g^{k}+f^{k-1}g^{k}-f^{k-1}g^{k-1}\,|\leq \]
\[ \leq |\,g^{k}\,||\,f^k-f^{k-1}\,|+|\,f^{k-1}\,||\,g^k-g^{k-1}\,|.\]
Then we obtain
\[ \|\,u (t)-\vartheta(t)\,\|\leq \bigg\{\|\,P(t)\,\|+\|\,Q(t)\,\|\Big[M_H+p\Big(M_{V_{1}}+M_{V_{2}}\Big) \Big]\bigg\} \times\]
\[\times L_{F_{u}}\|\,u(t)-\vartheta (t)\,\|+TM_{F}L_{F_{u}}\|\,u(t)-\vartheta (t)\,\|+TM_{F_u}L_{F}\|\,u(t)-\vartheta (t)\,\| \leq \]
\begin{equation} \label{Grind35}
\|\,u (t)-\vartheta(t)\,\| \leq \rho_{1} \|\,u (t)-\vartheta(t)\,\|,
\end{equation}
where
\[\rho_1=M_0L_{F_{u}}+TM_{F}L_{F_{u}}+TM_{F_u}L_{F},\:\;M_0=\|\,P(t)\,\|+\|\,Q(t)\,\|\Big[M_H+p\Big(M_{V_{1}}+M_{V_{2}}\Big) \Big].\]

Consequently, from the estimates (\ref{Grind34}) and (\ref{Grind35}) follows that the functional equation (\ref{Grind33}) has a unique solution $u(t) \in C\big([0,T],\mathbb{R}^n\big)$.
\end{proof}

One can solve the equations \eqref{Grind10} and \eqref{Grind33} as a one system and obtain control function $u(t)$ and state function $x(t)$.

{\centering
\section*{Conclusion}}

In the segment  $[0,T]$ we consider a nonlinear functional-differential equation with nonlinear function under the sign of first order differential. The functional-differential equation is considered with initial value conditions. To find control function in nonlinear problem, we use minimization of functional of quality.

For fixed values of control function we solve the nonlinear functional equation with initial conditions. For fixed values of state function we solve the nonlinear functional equation with final-point conditions.

The Pontryagin function is built and the condition of optimality is derived. In poving theorem we use the method of contracting mapping.

Moreover, the results obtained in this work will allow us in the future to investigate direct and inverse optimal control problems for other kind of functional-differential equations of mathematical physics with different quality functionals.


\noindent
\textbf{Data availability} This manuscript has no associated data.

\noindent
\textbf{Ethical Conduct} Not applicable.

\noindent
\textbf{Conflicts of interest}
The authors declare that there is
no conflict of interest.

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